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128

Jul 19, 2010
07/10

by
Zor Shekhtman

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The purpose of this lecture is to apply the general ideas of solving equations using invariant transformations to a particular case of linear equations. The most general form of linear equation is: A*x + B = 0 where coefficient A is not equal to 0. We will use invariant transformations to solve this equation: 1. Subtract B from both sides of an equation getting A*x = -B 2. Divide both sides of an equation by A (this is an invariant transformation since A is not equal to 0) getting x = -B/A....

Topics: Unizor, Math, Math4teens, Algebra, Linear, Equations, Geometry, Graph, Number

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93

Jul 21, 2010
07/10

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Zor Shekhtman

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The purpose of this lecture is to generalize a concept of an equation. The basis for this generalization is a concept of a function. We all know that x^2 = 1 is an equation and two real numbers 1 and -1 are its solutions. Based on this example, let's approach an "equation" from more formal position. First of all, if we speak about equation, we have to have some function on the left side of it. In the example above it was a function x2. Generally speaking, any function can participate...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number

29
29

Jul 22, 2010
07/10

by
Zor Shekhtman

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Using invariant and not invariant transformations, solve the following equations. Beware of loosing solutions as you apply certain not invariant transformations. X^2 - 16 = 0 (X^2 - 16)/((X + 4)^2) = 0 (X^2 - 1)*(X - 3) + X + 1 = 0 Notes: Transformation T(Y): Y â†’ âˆšY (where square root is understood as the positive only value) is not invariant; we loose the negative value of a square root. Transformation T(Y): Y â†’ Y*K is invariant only if K is not equal to 0. Therefore, if we multiply both...

Topics: Unizor, Math, Math4teens, Algebra, Quadratic, Equations, Problem, Solution, Geometry, Graph, Number

47
47

Jul 23, 2010
07/10

by
Zor Shekhtman

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Solve the following equation in the domain of real and complex numbers. X^3 + X^2 + X + 1 = 0 There are two different solutions to this equation offered on a video. One uses only invariant transformation of factoring out an expression and using the fact that if a product of two expressions is equal to 0 then either one could be zero. Another method is less trivial and uses non-invariant transformation but, in a subjective opinion of the author, is better from the esthetical viewpoint.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

356
356

Jul 26, 2010
07/10

by
Zor Shekhtman

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In this lecture we will consider different transformations that can be done with equations. Some of them will be invariant and will transform original equations into equivalent ones, so solving a transformed equation produces the same solutions as solving an original one. But there might be some other cases when we can use non-invariant transformations and have to be very careful applying transformed equations' solutions for original ones. We will always assume that we are attempting to solve...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

130
130

Jul 26, 2010
07/10

by
Zor Shekhtman

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We have expanded our choice of numbers from natural (1, 2, 3 etc.) to integer (adding 0 and negative numbers), to rational (adding fractions), to real (adding irrational numbers like square root of 2). Every expansion we made was initiated by a desire to do something that seemed to be reasonable but impossible to accomplish in the old framework. The last step on this way is introduction of complex numbers which will open one more previously closed door - extracting roots of any even degree from...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Function, Graph,...

37
37

Jul 26, 2010
07/10

by
Zor Shekhtman

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Complex Numbers - Problem 1 Express a square root from i in a standard complex form: a*i+b In other words, find real numbers a, b such that: (a*i+b)^2 = i

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Complex,...

47
47

Jul 27, 2010
07/10

by
Zor Shekhtman

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Complex Numbers - Problem 5 Express in a normal complex form x + y*i the result of a division of one complex number a + b*i by another c + d*i.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Function, Graph,...

33
33

Jul 24, 2010
07/10

by
Zor Shekhtman

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Prove the following identity: (X+Y)^4+X^4+Y^4 = 2*(X^2+X*Y+Y^2)2 It is obviously possible to open all the brackets, multiply (X+Y) by itself four times, square the expression on the right and come up with an identity. But this is quite boring. Try to think about a simpler way to prove this identity. One important guess will significantly shorten the proof. Finding this trick is the real subject of this problem.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

138
138

Jul 26, 2010
07/10

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Zor Shekhtman

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Here we will talk only about graphical representation of functions of one argument defined in the domain of real numbers and taking values in the same set of real numbers. For example, function y = x3 + 2x - 1. Functions deal with numbers but graphs are geometrical figures and, first of all, we have to define a correspondence between algebraic concept of a number and geometrical lines, points, curves etc. First of all, we define a system of Cartesian coordinates. We assume, there are two...

Topics: Unizor, Math, Math4teens, Algebra, Quadratic, Equations, Problem, Solution, Geometry, Graph, Number

30
30

Jul 26, 2010
07/10

by
Zor Shekhtman

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Complex Numbers - Problem 3 Solve in complex numbers an equation x^2 + 4x + 5 = 0

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Complex,...

39
39

Jul 21, 2010
07/10

by
Zor Shekhtman

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Solve the following quadratic equations in the domain of real numbers by invariantly transforming them into full square equations of type (X - A)^2 = B. 2*X^2 - 5*X - 3 = 0 4*X^2 + 12*X + 9 = 0 3*X^2 - 18*X + 30 = 0 Illustrate the solutions graphically.

Topics: Unizor, Math, Math4teens, Algebra, Quadratic, Equations, Problem, Solution, Geometry, Graph, Number

25
25

Jul 26, 2010
07/10

by
Zor Shekhtman

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Represent graphically the following function for different values of power r: y = x^R Consider real positive and negative, small and large values of power r.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Function, Graph,...

28
28

Jul 22, 2010
07/10

by
Zor Shekhtman

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Using the language of invariant transformations, solve equations: 2*(X + 4) - 10 = 0 (X + 4)/2 - 10 = 0 (X + 4)/2 - 10*(X + 1) = 0 Note: Transformations T(X): X â†’ X + a, T(X): X â†’ X * a (for a not 0) and T(X): X â†’ X / a (for a not 0) are invariant and cannot add wrong solutions to an original equation or loose real solutions. So, it is admissible to use solutions to a transformed equation as solutions to an original one. In theory, checking solutions in case only invariant transformations...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

36
36

Jul 26, 2010
07/10

by
Zor Shekhtman

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Represent graphically the following functions: y = |x| y = |x-1| + |x+1|

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

66
66

Jul 26, 2010
07/10

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Zor Shekhtman

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Complex Numbers - Problem 2 Consider a complex number represented graphically on a plane, i.e. if this number's algebraic representation is a + b*i then its graphical representation is a point A on a plane with coordinates (a, b). Let's further consider that this point (a, b) is located on the distance 1 from the beginning of coordinates (i.e. the length of 0A is equal to 1). Prove that (1) the square of this number represented graphically by a point B on the plane is located on the same...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Complex,...

106
106

Jul 28, 2010
07/10

by
Zor Shekhtman

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We have expanded a set of numbers we deal with from natural to integer to rational. Every time this expansion was caused by something mathematicians wanted to do with existing numbers but could not because the result of a reverse operation was not always defined in the existing set of numbers. Thus, zero and negative numbers had to be added to natural ones to enable to define unrestricted subtraction. Rational numbers had to be added to enable division. Now consider an operation of squaring a...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Complex,...

38
38

Jul 26, 2010
07/10

by
Zor Shekhtman

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Represent graphically the following function: y = |2*x+4| + |3*x-3| + |2-x|

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Function, Graph,...

19
19

Jul 22, 2010
07/10

by
Zor Shekhtman

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Causiously using invariant and not invariant transformations, solve equations: 2*âˆš(X + 4) - 10 = 0 (âˆš(X + 4))/2 - 10 = 0 (âˆš(X + 4))/2 - 10*âˆš(X + 1) = 0 where âˆš() is a square root function.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

25
25

Jul 25, 2010
07/10

by
Zor Shekhtman

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Prove that if A + B + C = 0 then A^3+B^3+C^3 = 3*A*B*C

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

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97

Jul 26, 2010
07/10

by
Zor Shekhtman

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What happens with a graph if we multiply an argument of a function? In other words, how will a graph of a function y = f(k*x) look? Using similar approach, if a point (A,B) belongs to an original graph (i.e. B = f(A)) then (A/k,B) belongs to a graph of a new function. Indeed, let's substitute x = A/k and y = B to a new function: f(k*(A/k)) = f(A) = B which proves that (A/k,B) belongs to a graph of a new function. Geometrically, the fact that for each point (A,B) of an original graph there is a...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

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30

Jul 26, 2010
07/10

by
Zor Shekhtman

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Complex Numbers - Problem 4 Prove that two roots of any quadratic equation are either both real or both complex with equal real parts and imaginary parts having equal absolute values but opposite signs. In other words, if one solution can be represented as a+b*i then another is a-b*i.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Complex,...

38
38

Jul 21, 2010
07/10

by
Zor Shekhtman

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Solve the following quadratic equations in the domain of complex numbers by invariantly transforming them into full square equations of type (X - A)^2 = B. 9*X^2 + 12*X - 12 = 0 25*X^2 + 70*X + 49 = 0 16*X^2 + 24*X + 57 = 0 Illustrate the solutions graphically.

Topics: Unizor, Math, Math4teens, Algebra, Quadratic, Equations, Problem, Solution, Geometry, Graph, Number

29
29

Jul 29, 2010
07/10

by
Zor Shekhtman

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Consider representation of a rational number as a periodic decimal fraction. Prove that product of two periodical decimal fractions with, generally speaking, different number of digits in a period is itself a periodic decimal fraction.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Function, Graph,...

45
45

Jul 26, 2010
07/10

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Zor Shekhtman

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1. What happens with a graph of a function y = f(x) if you add a constant a to argument x in its algebraic expression? In other words, where would be a graph of a function y = f(x + a)? Consider both cases a > 0 and a < 0. 2. What happens with a graph of a function y = f(x) if you add a constant a to the right side of its algebraic expression? In other words, where would be a graph of a function y = f(x) + a? Consider both cases a > 0 and a < 0. 3. What happens with a graph of a...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Function,...

33
33

Jul 28, 2010
07/10

by
Zor Shekhtman

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We can always consider a complex number a + b*i as a pair of two real numbers (a, b) and each such pair (i.e. each complex number) we can put into a correspondence with a point on a coordinate plane with coordinates (a, b). This seemingly simple one-to-one correspondence has far reaching ramifications. Plainly speaking, almost an entire subject of geometry on a plane can be reduced to studying complex numbers and their properties.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Complex,...

29
29

Aug 4, 2010
08/10

by
Zor Shekhtman

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Prove using the method of mathematical induction that N straight lines on a plain cross in N*(N-1)/2 points provided there are no parallel lines among them and no points where more than 2 lines cross each other.

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

115
115

Jul 29, 2010
07/10

by
Zor Shekhtman

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The concept of a function includes three major components: a set called "domain", a set called "codomain" and a rule which for each element of a domain points to (puts into a correspondence) a one and only one element from a codomain. Elements of a domain are called "arguments" of this function and corresponding elements of a codomain pointed to by an above mentioned rule are called "values" or "images" of this function. Symbolically, we can...

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

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123

Jul 29, 2010
07/10

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Zor Shekhtman

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Once people came to an abstract idea of a number, they had to write it in some way. Probably, the most primitive numerical system is when a number is represented by a corresponding set of vertical bars. Thus, number "five" is represented by a sequence of five bars "|||||". Imagine now how difficult it was for cavemen to write a contract about a sale of, say, 38 sheep. Help could have come in a form of a grouping. Assume, we reserve a symbol 'A' to designate a number...

Topics: Unizor, Math, Math4teens, Lecture, Algebra, Geometry, Graph, Invariant, Transformation, Equations,...

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57

Jul 29, 2010
07/10

by
Zor Shekhtman

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Let's prove the converse theorem that any rational number can be represented by a finite or infinite periodic decimal string. Obviously, the whole proof is a logical reverse of a proof that any periodic decimal fraction is a rational number which was presented in the previous lecture. I will make a few simplifications of this problem to prove it in a relatively simple case from which the general case will follow trivially. My first simplification is to reduce the proof to only rational numbers...

Topics: Unizor, Math, Math4teens, Lecture, Algebra, Geometry, Graph, Invariant, Transformation, Equations,...

41
41

Aug 4, 2010
08/10

by
Zor Shekhtman

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Prove using the method of mathematical induction that for all natural N > 3 the following inequality holds: N! > 2N where N! is 1*2*...*(N-1)*N - product of all natural numbers from 1 to N.

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

42
42

Jul 29, 2010
07/10

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Zor Shekhtman

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In the previous lecture we have attempted to define rational numbers more rigorously using formalism of strings of a form {M|N} where M and N are integer numbers (N is not 0). We have also mapped all integer numbers into a subset of these strings so that integer number K corresponds to a rational number {K|1}. If, discussing rational numbers of a form {M|N}, we mention integer number K, it just means that we are considering its corresponding rational equivalent {K|1} and use K as a shorthand...

Topics: Unizor, Math, Math4teens, Lecture, Algebra, Geometry, Graph, Invariant, Transformation, Equations,...

49
49

Aug 4, 2010
08/10

by
Zor Shekhtman

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Arithmetic average of N numbers is (X1+X2+...+XN) / N. Geometric average of these N numbers is (X1*X2*...*XN)^1/N (i.e. an Nth root of their product). Prove using the method of mathematical induction that arithmetic average of N positive numbers is greater or equal than their geometric average. This inequality is often attributed to French mathematician Augustin Cauchy.

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

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95

Jul 29, 2010
07/10

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Zor Shekhtman

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What is a decimal representation of a number? It is a finite or infinite sequence (string) of decimal digits with optional sign in front and optional separator between integer and fractional (if it exists) parts - a dot or a comma depending on a country standard. If there is no sign in front of this string we assume a sign '+' (plus) is present. If there is no separator we assume it is after the right most digit in a string. So, any number (integer or fractional) in decimal system can be...

Topics: Unizor, Math, Math4teens, Lecture, Algebra, Geometry, Graph, Invariant, Transformation, Equations,...

107
107

Jul 29, 2010
07/10

by
Zor Shekhtman

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We start with an assumption that integer numbers are known, and an operation of multiplication among them is well defined. We also assume that sometimes a reverse operation of division is also defined for two integer numbers (e.g. 15 / 5 = 3) but it is defined only for some pairs (if, as we say, one integer is divisible by another). We would like to expand a concept of integer numbers to make an operation of division possible for any pair of integers. This lecture introduces a formal and...

Topics: Unizor, Math, Math4teens, Lecture, Algebra, Geometry, Graph, Invariant, Transformation, Equations,...

29
29

Jul 29, 2010
07/10

by
Zor Shekhtman

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Consider representation of a rational number as a periodic decimal fraction. Prove that sum of two periodical decimal fractions with, generally speaking, different number of digits in a period is itself a periodic decimal fraction.

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

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75

Jul 28, 2010
07/10

by
Zor Shekhtman

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I'd like to approach rational numbers from a familiar concepts of harmony, symmetry and groups addressed previously. Our basis will be a set of natural numbers 1, 2, 3, etc. After successfully introducing an operation of addition mathematicians came up with a concept of multiplication. Now they could replace a repetitive additions with a single operation of multiplication. Again, together with any new operation the principle of harmony and symmetry requires existence of a unit element which,...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

49
49

Aug 2, 2010
08/10

by
Zor Shekhtman

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Prove using the method of mathematical induction that the sum of the first N odd numbers is N^2.

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

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34

Jul 28, 2010
07/10

by
Zor Shekhtman

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We start with an assumption that natural numbers 1, 2, etc. are known, order among them is established (i.e. we know which one follows the other and is greater than the previous) and an operation of addition among them is well defined. So, for every pair of natural numbers (say, 5 and 8) an operation of addition (denoted by a plus sign between these two numbers) is defined and result of this operation of addition in a known natural number (5 + 8 = 13). We also assume that sometimes an operation...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

30
30

Aug 4, 2010
08/10

by
Zor Shekhtman

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Prove using the method of mathematical induction that the sum of internal angles of convex polygon with N sides is equal to 180*(N-2) degrees.

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

158
158

Aug 2, 2010
08/10

by
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Mathematical induction can be called upon when we want to prove that some formula or a rule is true for all natural numbers. It is done in three steps. Firstly, we check that the formula or a rule holds for one particular natural number. For example, the formula 1+2+3+...+(N-1)+N = N*(N+1)/2 should be checked for N = 1. Indeed, it holds since the left side is a sum from 1 to 1, i.e. 1; the right side is 1*(1+1)/2 = 1. Checked. Secondly, we assume that a formula or a rule holds for some...

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

177
177

Aug 5, 2010
08/10

by
Zor Shekhtman

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Sets, numbers, geometrical figures, equations etc. are what mathematicians are talking about. Logic is their language. You cannot talk about any mathematical property without using the terminology of logic. There are a few quite fundamental logical concepts which everybody who wants to talk about math needs to know. This is a brief introduction to these concepts. TRUE or FALSE - this is the question most mathematicians are concern about. They express certain statement about some properties of...

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

37
37

Aug 4, 2010
08/10

by
Zor Shekhtman

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Prove the binomial formula: (a+b)^n = C(n,0)*a^n+C(n,1)*a^(n-1)*b^1+...+C(n,n-1)*a^1*b^(n-1)+C(n,n)*b^n where C(n,k) is n!/(k!*(n-k)!).

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

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50

Aug 5, 2010
08/10

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There is an important terminology for logical conditions and their relationship when dealing with proof of theorems. It is important to be familiar with it to properly understand what exactly a theorem states. Usually, theorem states something like "If [some condition is true] then [some other condition is true]". Equally acceptable formulation is "From [some condition is true] follows [some other condition is true]". If from a statement A (e.g. some condition is true)...

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

105
105

Jul 28, 2010
07/10

by
Zor Shekhtman

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Let's go back in history by many thousands of years when people just learned to add positive integer numbers. Five sheep and eight sheep is thirteen sheep. Five bulls and eight bulls is thirteen bulls. Five sheep and eight bulls is thirteen animals. Stop! This is something new. What we are witnessing here is a major shift from numbers representing concrete objects we can count to numbers as mathematical abstractions. No longer we have to learn how to add sheep separately from how to add bulls,...

Topics: Unizor, Math, Math4teens, Algebra, Invariant, Transformation, Equations, Geometry, Graph, Number,...

46
46

Aug 5, 2010
08/10

by
Zor Shekhtman

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Given a set: book, pen, football, building, planet. How many elements does this set have? What are they? What subsets, including trivial, does this set have? What elements belong to a subset of "man-made things"?

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

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41

Aug 5, 2010
08/10

by
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Given two sets. Set A contains elements: book, pen, football, building, planet. Set B contains elements: football, golf ball, tennis ball. What set is a result of an operation of intersection between sets A and B? What set is a result of an operation of union between sets A and B? What is a result of a union between set A and an empty set? What is a result of an intersection between set A and an empty set? What is the smallest (in number of elements) subset of a set A?

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...

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33

Aug 5, 2010
08/10

by
Zor Shekhtman

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Given a set of N different elements. Let's say, elements of this set are numbers from 1 to N. How many subsets does this set have, including trivial ones (empty subset and a subset equal to an entire set)?

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Aug 5, 2010
08/10

by
Zor Shekhtman

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Set theory is an independent mathematical subject, well developed and used. We will use just certain concepts of set theory and utilize its language. Sets and their elements are elementary objects which we can only describe and exemplify. Thus, a set of all people who live on this planet includes elements like you and me. A set of all points on a plane can contain a subset of points that are inside of a given circle on this plane. Sets, like numbers, can be operated upon. As with adding two...

Topics: Unizor, Math, Math4teens, Lecture, Function, Algebra, Geometry, Graph, Invariant, Transformation,...